It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links.
The Morton–Franks–Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type.
We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for and then it is also true for the –cable of and for the connect sum of and
"The algebraic crossing number and the braid index of knots and links." Algebr. Geom. Topol. 6 (5) 2313 - 2350, 2006. https://doi.org/10.2140/agt.2006.6.2313